Ch. 3 - Describing, Exploring, and Comparing Data

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 3 - Describing, Exploring, and Comparing Data

Problem 3.1.2

Chapter 3, Problem 3.1.2

What’s Wrong? Education Week magazine published a list consisting of the mean teacher salary in each of the 50 states for a recent year. If we add the 50 means and then divide by 50, we get \$56,479. Is the value of \$56,479 the mean teacher salary for the population of all teachers in the 50 United States? Why or why not?

Verified step by step guidance

Step 1: Understand the concept of a mean. The mean is calculated by summing all values in a dataset and dividing by the number of values. In this case, the dataset consists of the mean teacher salaries for each state.

Step 2: Recognize the distinction between the mean of means and the overall population mean. The value \$56,479 is the mean of the state-level means, not the mean of all individual teacher salaries across the United States.

Step 3: Explain why the mean of means may not represent the population mean. Each state has a different number of teachers, so the state-level means are weighted differently in the overall population mean. States with more teachers contribute more to the population mean than states with fewer teachers.

Step 4: To calculate the true population mean, you would need to sum the salaries of all individual teachers across the 50 states and divide by the total number of teachers. This approach accounts for the varying number of teachers in each state.

Step 5: Conclude that \$56,479 is not the mean teacher salary for the population of all teachers in the United States. It is simply the mean of the state-level means, which does not account for the differing teacher populations in each state.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mean vs. Weighted Mean

The mean is calculated by summing all values and dividing by the number of values. However, in this context, the mean teacher salary for each state may not represent the overall mean for all teachers if the number of teachers varies significantly between states. A weighted mean, which accounts for the number of teachers in each state, would provide a more accurate representation of the average salary across the entire population.

Population vs. Sample

In statistics, a population includes all members of a defined group, while a sample is a subset of that population. The mean salary calculated from the state means does not consider the distribution of teachers within each state, making it a sample mean rather than a true population mean. This distinction is crucial for understanding the limitations of the reported average.

Variability and Distribution

Variability refers to how much the data points differ from each other and from the mean. In this case, teacher salaries can vary widely between states due to factors like cost of living and funding. Understanding the distribution of salaries is essential, as it affects the interpretation of the mean and whether it accurately reflects the salary landscape for all teachers.

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